Fluid dynamics in intracranial aneurysms treated with flow-diverting stents: effect of multiple geometrical parameters

Abstract

Characterizing the haemodynamics in intracranial aneurysms is of high interest as it impacts aneurysm growth, rupture and treatment, especially with flow-diverting stents (FDS). Flow in these geometries is known to depend on the Dean, Reynolds and Womersley numbers, $De,Re,Wo$ , but is also influenced by geometrical parameters such as the sac shape or the size of the opening. Via particle image velocimetry, this parametric study aimed at evaluating the combined effects of $Re$ , $De,Wo$ and the geometry of the aneurysmal sac on the haemodynamics before and after treatment with FDS. Eight ellipsoidal idealized aneurysm models were created with two curvatures of the parent vessel, two aspect ratios of the sac and two neck sizes. Before treatment, a single counter-rotating vortex, whose strength increases with $Re$ and $De$ , as well as with the neck size and the aspect ratio, was observed in the sac for all but one geometry. After treatment with FDS, four different flow topologies were observed, depending on the geometry: no separation, separation for part of the cycle, two opposing vortices or a single counter-rotating vortex. A linear model with interaction revealed the predominant effect of $De$ and the curvature of the parent vessel on the haemodynamics before and after treatment. This work once more demonstrated the primary role of haemodynamics in the treatment of intracranial aneurysms with FDS. Future work will consider the complexity of patient-specific geometries, and their effects on both the haemodynamics in the sac and the porosity of the FDS.

Keywords: biomedical flows; blood flow.

Figure 1.

(a) Illustration of the idealized aneurysm geometry and the different geometrical parameters. ((b) Four idealized aneurysm geometries $\left(\kappa =0.22{\mathrm{m}\mathrm{m}}^{-1}\right)$ with different aspect ratios ($AR$) and neck sizes ($D_{neck}$).

Figure 2.

Velocity fields $\mathit{u}$ for four geometries for ${\overline{Re}}_{PV}=414$ at four time points of the cardiac cycle: A (yellow), the flow in the parent vessel accelerates; B (red), the flow rate in the parent vessel is maximum; C (green), the flow in the parent vessel decelerates; D (blue), the flow rate in the parent vessel is minimum. The background colour represents the out-of-plane velocity and the colour of the arrows represents the magnitude of the in-plane velocity.

Figure 3.

Location of the centre of the vortex for 20 time points of the cycle and for (a–c) three different aneurysm geometries, for ${\overline{Re}}_{PV}=414$ (i.e. $Q_{PV}=300\mathrm{m}\mathrm{l}{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$).

Figure 4.

Velocity profile at the neck for $\mathrm{K}-/\mathrm{A}\mathrm{R}+/\mathrm{N}-$ (a) and $\mathrm{K}-/\mathrm{A}\mathrm{R}-/\mathrm{N}-$ (b) during the acceleration phase (start of the cardiac cycle to peak systole) in the parent vessel (line becoming darker as the velocity in the parent vessel increases) for ${\overline{Re}}_{PV}=414$ (i.e. $Q_{PV}=300\mathrm{m}\mathrm{l}{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$) and $Wo=2.5$. Time-averaged $Re$ at the neck $\left(R{e}_N\right)$ as a function of $Re$ in the parent vessel $\left(R{e}_{PV}\right)$ for aneurysmal sac geometries with large (c) and small (d) aspect ratio.

Figure 5.

Time-averaged circulation in the sac, $⟨\Gamma ⟩$, as a function of $R{e}_{PV}$ for all curvatures and neck sizes for high (a) and low (b) $AR$. (c) Reynolds number in the aneurysm $R{e}_A$ for the large neck size ($D_{neck}=5\mathrm{m}\mathrm{m}$) as a function of $R{e}_A$ for the small neck size $\left(D_{neck}=3\mathrm{m}\mathrm{m}\right)$, for two frequencies of the flow in the parent vessel: 50 beats per minute in solid line and 100 beats per minute in dashed line ($Wo=2.5$ and $Wo=3.2$).

Figure 6.

(a) Time-averaged circulation in the sac $⟨\Gamma ⟩$ normalized by the kinematic viscosity $\nu$ as a function of $De$ for the lowest value of the sac $AR$ and two different neck sizes (combining all curvature conditions). (b) Time-averaged circulation in the sac $⟨\Gamma ⟩$ normalized by the kinematic viscosity $\nu$ as a function of $De$ for $D_{neck}=5\mathrm{m}\mathrm{m}$ and three aspect ratios ($AR=1$ corresponding to a spherical shape).

Figure 7.

Velocity fields $\mathit{u}$ after treatment with FDS for four geometries and for ${\overline{Re}}_{PV}=414$ at four time points of the cycle: A (yellow), the flow in the parent vessel accelerates; B (red), the flow rate in the parent vessel is maximum; C (green), the flow in the parent vessel decelerates; D (blue), the flow rate in the parent vessel is minimum. The background colour represents the out-of-plane velocity and the grey scale of the arrows represents the magnitude of the in-plane velocity.

Figure 8.

Velocity profile at the neck for $\mathrm{K}-/\mathrm{A}\mathrm{R}+/\mathrm{N}+$ (a), $\mathrm{K}-/\mathrm{A}\mathrm{R}-/\mathrm{N}+$ (b), $\mathrm{K}+/\mathrm{A}\mathrm{R}+/\mathrm{N}+$ (c) and $\mathrm{K}+/\mathrm{A}\mathrm{R}+/\mathrm{N}-$ (d) during the acceleration phase of the flow in the parent vessel (line becoming darker as the velocity in the parent vessel increases) for ${\overline{Re}}_{PV}=414$ (i.e. $Q_{PV}=300\mathrm{m}\mathrm{l}{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$).

Figure 9.

(a) Normalized time-averaged circulation in the sac across the cardiac cycle period for $\mathrm{K}-/\mathrm{A}\mathrm{R}-/\mathrm{N}+$ and for three values of ${\overline{Re}}_{PV}$, as well as for $Wo=2.5$ and $Wo=3.2$. (b) Time-averaged circulation in the sac as a function of $De$. The marker shape corresponds to the sac $AR$, triangle for $\mathrm{A}\mathrm{R}-$ and circle for $\mathrm{A}\mathrm{R}+$, and the colour corresponds to the flow topology in the sac.

Figure 10.

Percentage of the variation of $⟨\bar{u}⟩$ induced by the three parameters $De,D_{neck}$ and $AR$ and their interactions before (in black) and after (in grey) treatment with FDS.

Figure 11.

(a) Ratio between the magnitude of the time-averaged circulation $⟨\left|\Gamma \right|⟩$ after and before FDS treatment, as a function of $De$, for the three aspect ratios. (b) Effect of the three parameters $De,D_{neck}$ and $AR$ and their interactions on the reduction of the magnitude of the time-averaged circulation ($R^2=0.55$).

Figure 12.

(a) Stent geometries for two aneurysm models reconstructed from synchrotron X-ray micro-tomography images. (b) Distribution of the porosity at the neck for three different models, highlighting the variation of porosity with neck size $D_{neck}$ and curvature $\kappa$ of the parent vessel.